Subnet (mathematics)

If ${\displaystyle x_{\bullet }=\left(x_{a}\right)_{a\in A}}$ and ${\displaystyle y_{\bullet }=\left(y_{b}\right)_{b\in B}}$ are nets from directed sets A and B respectively, then ${\displaystyle y_{\bullet }}$ is a subnet of ${\displaystyle x_{\bullet }}$ if there exists a monotone final function

${\displaystyle h:B\to A}$

such that

${\displaystyle y_{b}=x_{h(b)}}$ for all ${\displaystyle b\in B.}$

A function ${\displaystyle h:B\to A}$ is monotone if ${\displaystyle b\leq c}$ imples ${\displaystyle h(b)\leq h(c)}$ and it is called final if its image is cofinal in A—that is, for every ${\displaystyle a\in A}$ there exists a ${\displaystyle b\in B}$ such that ${\displaystyle h(b)\geq a.}$[note 1]

The definition generalizes some key theorems about subsequences:

• A net ${\displaystyle x_{\bullet }}$ converges to x if and only if every subnet of ${\displaystyle x_{\bullet }}$ converges to x.
• A net ${\displaystyle x_{\bullet }}$ has a cluster point y if and only if it has a subnet ${\displaystyle y_{\bullet }}$ that converges to y.
• A topological space X is compact if and only if every net in X has a convergent subnet (see net for a proof).

A seemingly more natural definition of a subnet would be to require B to be a cofinal subset of A and that h be the identity map. This concept, known as a cofinal subnet, turns out to be inadequate. For example, the second theorem above fails for the Tychonoff plank if we restrict ourselves to cofinal subnets.

While a sequence is a net, a sequence has subnets that are not subsequences. For example the net (1, 1, 2, 3, 4, ...) is a subnet of the net (1, 2, 3, 4, ...). The key difference is that subnets can use the same point in the net multiple times and the indexing set of the subnet can have much larger cardinality. Using the more general definition where we don't require monotonicity, a sequence is a subnet of a given sequence, if and only if it can be obtained from some subsequence by repeating its terms and reordering them.[1]

• Filters in topology#Subnets

• Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3885380064.
• Kelley, John L. (1991). General Topology. Springer. ISBN 3540901256.
• Runde, Volker (2005). A Taste of Topology. Springer. ISBN 978-0387-25790-7.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0486434796.